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Topic: Is this series uniformly convergent for x != 0 ?
Replies: 5   Last Post: Feb 11, 2013 5:28 AM

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quasi

Posts: 10,197
Registered: 7/15/05
Re: Is this series uniformly convergent for x != 0 ?
Posted: Feb 11, 2013 2:24 AM
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quasi wrote:
>quasi wrote:
>>vv wrote:
>>>
>>>I'd be grateful if someone can throw light on whether or not
>>>the following series is uniformly convergent for x not equal
>>>to zero:
>>>
>>>\sum_{n=1}^infty exp(-ixn)/n

>>
>>I could be wrong, but here's what I think ...
>>
>>If k is a nonzero integer then for x = 2*k*Pi, the series
>>diverges.
>>
>>More generally, I think the series diverges for x = (2*k*Pi)/d
>>where k,d are nonzero integers with d odd and with k,d
>>relatively prime. Thus, the series is pointwise divergent on a
>>dense subset of R, so the question of uniform convergence is
>>silly.
>>
>>In fact, going out on a limb, it seems to me that the series
>>diverges for all real numbers x except for x = 0, x = Pi,
>>x = -Pi.

>
>I meant: except for x = Pi, x = -Pi.


My last claim now seems to be too strong.

I think the series diverges for all x except for x = (k*Pi)/d
where k,d are integers with k odd and k,d relatively prime.

Thus, the series converges pointwise for only countably many
real values of x. Moreover, even on its countable domain, it
seem clear that the convergence is not uniform.

quasi



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